Integration using trapezoidal rule in python

Опубликовано: 17 Ноябрь 2024
на канале: ScriptGPT

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integration using the trapezoidal rule in python

the trapezoidal rule is a numerical method for estimating the definite integral of a function. it works by approximating the area under the curve as a series of trapezoids rather than using rectangles as in the simpler riemann sums.

#### how the trapezoidal rule works:

1. **divide the interval**: split the interval \([a, b]\) into \(n\) equal segments, each of width \(h = \frac{b - a}{n}\).

2. **evaluate the function**: calculate the value of the function at the endpoints and at each subdivided point.

3. **calculate the area**: the area under the curve is approximated by summing the areas of the trapezoids formed. the area of each trapezoid is given by:
\[
\text{area} = \frac{h}{2} \times (f(x_0) + f(x_1) + 2 \sum_{i=1}^{n-1} f(x_i))
\]
where \(f(x_0)\) and \(f(x_n)\) are the function values at the endpoints.

#### steps for implementation in python:

1. define the function you want to integrate.
2. implement the trapezoidal rule.
3. call the function with your desired limits and number of intervals.

example code

here’s a complete implementation of the trapezoidal rule in python:

explanation of the code:

**function definition**: the `trapezoidal_rule` function takes another function `func`, the limits of integration `a` and `b`, and the number of subintervals `n`.
**step size**: the width of each trapezoid is calculated.
**integral calculation**: it initializes the integral with the average of the function values at the endpoints and sums the values of the function at each interval.
**multiplication**: finally, it multiplies the accumulated sum by the step size to get the final integral value.

example output

when you run the above code, it will output:

this is the correct result for the integral of \(f(x) = x^2\) from \(0\) to \(1\), which is \(\frac{1}{3}\).

conclusion

the trapezoidal rule is a simple yet effective method for numerical integration. the accurac ...

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